1. Field of the Invention
The invention is directed toward systems for measuring materials properties such as strain, crystallinity, thickness, purity, composition, and the like of samples; or for observing structures in transparent materials; or, for measurement of strain in models constructed for that purpose. It is more narrowly directed toward measuring systems that utilize polarized light for such measurements. It may be used in applications including scientific research, industrial measurement, quality control, forensics, and medical imaging.
2. Description of the Related Art
Polarization interference is a well-known way to observe birefringence or retardance in a sample. Birefringence is an intensive property of a sample whereby light polarized along different axes will experience different indices of refraction. The axis along which the index is lowest is termed the fast axis, and that along which the index is highest is termed the slow axis; these are necessarily perpendicular to one another. The optical indices are termed nf and ns.
Retardance is an extensive quantity measuring the total optical path difference experienced in passing through a sample, so for a uniform sample at normal incidence
R=(nsxe2x88x92nf)*dxe2x80x83xe2x80x83[1]
where nf and ns are the optical indices for light polarized along the fast and slow axes, respectively, and d is the thickness of the sample. More generally, in a sample where the values of ns and nd may vary along the line of sight due to e.g. inhomogeneities, but the fast axis orientation is constant, R is the integral of index difference over distance.
When a birefringent object is viewed through parallel or crossed linear polarizers, a pattern of colored fringes is seen. If the orientation of the slow axis varies across the face of the part, the fringe pattern will change as the polarizers are rotated. Alternatively, one may illuminate the sample in left-hand circularly polarized light and view it with a left-hand circular polarizer. This eliminates the dependence on polarizer orientation, while preserving the location and color of the fringes. These colored fringes (which simply appear light and dark when viewed in monochromatic light) are termed the set of isochromes, the lines consisting of the locii of points that share a given value of xcex4.
Another set of crucial patterns is the isoclines, being the locii of points which are completely extinguished when the sample is viewed between crossed polarizers. The isoclines change when the polarizers are rotated relative to the sample, and indicate those points where the principal optical axis in the sample is parallel to one of the polarizers. The isoclines are often used to determine the crystal orientation, or a stress distribution, throughout a sample. But observations of this type alone do not reveal which is the fast axis and which the slow axis.
There is a large literature describing apparatus and methods for determining the retardance in a sample. These are typically based on the intensity of the interference pattern of the isochromes, given by:
I=I0 cos (xcex4)2xe2x80x83xe2x80x83[2a]
for a sample placed between parallel polarizers, or
I=I0 sin (xcex4)2xe2x80x83xe2x80x83[2b]
when between crossed polarizers, where
xcex4=xcfx80R/xcexxe2x80x83xe2x80x83[3]
and I0 is the intensity of the incident light, which is monochromatic with wavelength xcex. Polychromatic light may be analyzed as a sum or integral of various wavelength components.
Equations [2a] and [2b] do not specify a unique value of xcex4 for a given observed intensity I, because the sin( ) and cos( ) functions are periodic. Typically, these equations are solved to yield a value xcex40 in the range [0, xcfx80/2], which is related to the actual xcex4 by either
xcex4=mxcfx80+xcex40xe2x80x83xe2x80x83[4a]
or
xcex4=mxcfx80xe2x88x92xcex40xe2x80x83xe2x80x83[4b]
where the indeterminacy between [4a] and [4b ] arises from the fact that cos( )2 for either of these two arguments yields the same answer; this is also true of sin( )2. Simply measuring the pattern of isochromes between crossed or parallel polarizers does not provide enough information to specify which case applies, and combining crossed and parallel measurements gives no further data, since sin( )2 and cos( )2 are inherently complementary. Then there is the further indeterminacy of the order m. So, in attempting to relate a retardance to an observed intensity between polarizers, one must overcome the uncertainty in order, m, and also determine whether the sample is described by equation [4a] or [4b].
In analogy with the description of xcex40 as the apparent phase, one may speak of the apparent retardance R0 which is defined to lie in the range [0, xcex/2] and is related to the actual retardance R by
R=mxcex+R0xe2x80x83xe2x80x83[5a]
xe2x80x83R=mxcexxe2x88x92R0xe2x80x83xe2x80x83[5b]
Except when the actual retardance is known to be less than xcex/2, one must use [5a] or [5b] together with a determination of the order, m, to calculate the actual retardance.
Some hardware used for polarization interference measurements provides additional data through the use of additional sensors, polarizing elements, waveplates, photoelastic modulators, and the like.
Oldenbourg et. al. U.S. Pat. No. 5,521,705 teaches how to unambiguously identify the slow axis and value of xcex4 for a retarder. This method uses an imaging detector and variable retarders, but it only functions for xcex4 in the range [0, xcfx80/2]. It is unable to determine the order, m.
Mason U.S. Pat. No. 5,917,598 teaches how to identify the fast axis using circularly polarized illumination and multiple linear polarizing analyzers with different orientations. This system appears to resolve phase xcex40 over the range [0, xcfx80/2] for monochromatic light, or over a wider range of xcex40 for broad-band light analyzed with a spectrometer. The determination of order, m, is only possible when broad-band light is used, and comes from analysis of the spectral distribution. But the requirement for a spectrometer to analyze the content of light passing through the sample means that system is limited in practice to measuring a single point in the sample at a time.
Croizer et. al. U.S. Pat. No. 4,914,487 determine isochromic fringes and then calculate absolute retardance by presuming stress levels at the end points of the sample, and integrating spatially across the sample using a finite-element stress equation. Since stress is related to birefringence by the stress-optic tensor, the stress equation provides additional information about birefringence levels that is said to allow unambiguous determination of xcex4 from xcex40, provided that the birefringence in the sample is entirely due to stress rather than e.g. crystallinity or other internal structures. Further, it requires one to spatially oversample, i.e. have a pixel scale that is considerably finer than any of the structures present, in order to perform the integration accurately. In practice, this approach is slow because of the need for the calculation step, and has proven unreliable when applied to real-world samples.
Others have observed samples at multiple wavelengths in an attempt to determine retardance in excess of xcex/2.
Young uses a linear polarizer to illuminate a single point in a sample, which is analyzed by a quarter-wave plate whose angle is keyed to that of the entrance polarizer, followed by a final rotating linear polarizer used as an analyzer. This last element is rotated to seek maximum extinction, and the angle of maximum extinction is noted. This is performed while the sample is illuminated at two wavelengths in turn. From the combinations of the polarizer setting angles, retardances in excess of xcex/2 are identified. But the system is inherently point-wise in nature, and cannot be readily extended to produce an image of retardance at every point in an object, except by taking a grid of point-wise measurements. This is impractical when one seeks images having even moderate spatial definition. For example, a conventional video image has resolution of 640xc3x97480 pixels, thus contains 307,200 individual points, which would take in excess of a day to acquire at 1 second per point.
Robert et. al. U.S. Pat. No. 4,171,908 uses a rotating-polarizer system to make point measurements at two wavelengths together with another sensor that incorporates a quarter-wave plate whose orientation is servo-controlled, in order to determine the fast axis and the relative phase (xcfx862xe2x88x92xcfx861) between the two wavelengths. Phase-sensitive information from the polarizer and servo-control of the quarter-wave plate indicates the sense of the relative phase. While it is said to provide complete information about the sample retardance, this arrangement is complex and has many moving parts. Further, it measures only a point at a time, so is ill-suited to providing a high-definition image of birefringence or retardance across an entire sample. As with Young, when an image is desired, one must account low speed as an additional shortcoming, along with the expense, complexity, and size involved.
Sakai et. al. in U.S. Pat. No. 4,973,163 uses two wavelengths to resolve order by comparing all possible R that satisfy the observed value of R0(xcex1) against all possible R that satisfy the observed R0(xcex2), then choosing the value that is most compatible with both readings. Identification of fast and slow axes is not addressed in this teaching. Contrary to the teaching, however, this method does not provide an unambiguous determination of R, since there are in fact many cases where two different values of R yield the same set of readings {R0(xcex1), R0(xcex2)}. Examples are provided below in connection with the Stockley et. al. patent. Thus, Sakai et. al. does not actually provide for unambiguously determining retardance.
Sakai et. al. also teach use of three xcex and comparison amongst R0 obtained at each of them to arrive at a unique value of R consistent with all three, using a Cauchy fit to account for birefringence dispersion. This could remove the ambiguities inherent in the two wavelength approach, but requires taking a third set of measurements at a third wavelength, and even then would only work if one had a Cauchy fit to the material, which is often unknown. With this approach, one would face the situation that the optical properties of the sample material need to be known, before a sample can be measured.
In U.S. Pat. No. 5,400,131, Stockley et. al. teach use of measurements at wavelengths xcex1 and xcex2 together with a look-up table, to obtain a value of R from the two measured R0. However, this system suffers from the same ambiguities as the Sakai et. al. system, which are either not recognized or not acknowledged by the inventors. FIG. 6 of this patent clearly illustrates the shortcomings of the system. It is a graph with R0(xcex1) and R0(xcex2) as the x and y axes, on which a curve is plotted that indicates the value of R corresponding to the observed R0 coordinates. Yet whenever this curve crosses over itself, as it does many times in the Figure, there are two quite different values of R that fit the observations equally well, leading to ambiguous determination of R. On this crucial issue the teaching is silent. This is the same defect mentioned in connection with Sakai et. al. and it is equally limiting here.
Stockley et. al. also teach a system using three wavelengths that are chosen to fit a geometric series, i.e. xcex1*xcex3=(xcex2)2. Two wavelengths at a time are combined to yield a ramp function. From two such ramp functions, it is said to be possible to resolve the actual retardance from the apparent retardance values R0. The output of the ramp processing is a retardance map that is periodic over the range [0, xcfx80] and has discontinuities at the boundaries corresponding to xcex4=mxcfx80. These must be detected and fixed by spatial analysis of the structure being imaged, in which a computer stitches together an image that resolves the proper order m for each region based on the observed discontinuities and the overall topology. This suffers the same problems as the Croizer approach described above, since it requires computation, and there must be significant spatial oversampling and/or certain assumptions must be made about the structure being imaged.
By extending the observation from two or three wavelengths to a complete spectrum, Mason in U.S. Pat. No. 5,825,493 teaches measurement of retardance in excess of xcex/2. The hardware uses a broad-band source for illumination and a spectrometer for analysis. For each point to be imaged, a spectrum is acquired and compared to a reference spectrum obtained with no sample present; from the spectral ratio, the retardance is determined. This suffers from the same limitations as the other system of Mason, namely the need for a spectrometer which makes it inherently a point sensing system that cannot readily be used to produce a two-dimensional image of a sample.
In summary, while there is extensive art for obtaining and interpreting isochrome and isocline images, none are satisfactory for providing identification of the fast and slow axis together with unambiguous determination of R in excess of xcex/2 across a complete image. All methods that seek to provide such information suffer from one or more of the following limitations: need for spatial analysis of the image to resolve order; reliance on stress-integral analysis to resolve order and provide axis determination; need for a priori knowledge about the distribution of stress and/or birefringence in the sample; possibility for confusion of certain values of retardance with other, widely-different values of retardance; need for observations at three wavelengths or across a continuous spectrum; inherent limitations that in theory permit measuring at most a line at a time, and in practice, a single point at a time.
Thus there is no system presently known for measuring retardance in excess of xcex/2 that is reliable, that provides a high-definition image of the sample, that identifies the fast and slow axis orientation, that needs no predetermined information about the sample, and does not rely upon spatial distribution in the sample to resolve order from apparent phase xcex40 or apparent retardance R0.
It is a goal of the present invention to obtain images of retardance in a sample where R can exceed xcex/2, and more typically can be 5xcex or more. It is another goal to provide a determination of R that is independent for each point, with no need for spatial analysis or a priori knowledge of the sample. A further goal is to eliminate the need for taking spectra, and to provide a high-definition image of the entire sample at once.
Yet another object of the present invention is to eliminate ambiguities and misidentifications of R, and to enable any sample to be measured without prior knowledge such as a Cauchy dispersion fit to birefringence. To the contrary, rather than needing this information as an input, it is an aim of the present invention to provide a determination of birefringence dispersion in the sample as part of the measurement.
The invention is based on the recognition that, when observing at a wavelength band centered at xcex1, a retarder having R=xcex1 and a fast axis oriented at angle xcex2 is indistinguishable from a retarder having R=mxcex1xe2x88x92xcex1 and a fast axis oriented at xcex2+xcfx80/2. Similarly, when observed at xcex2, the retarder may appear to have R=mxcex2xe2x88x92xcex1 and a fast axis oriented at angle xcex2+xcfx80/2. Yet, in a real sample, the fast axis does not vary with the wavelength of observation. This provides an additional constraint on the problem which may be used to eliminate the ambiguities that are present in prior art approaches.
At its simplest level, one may use hardware such as that of Oldenbourg to obtain an image of R0 in the range [0, xcfx80/2] and to identify the fast axis orientation at each point in a sample, at each of two wavelengths xcex1 and xcex2. Then, LUTs are used to convert apparent retardance R0 to actual retardance R, as in Stockley et. al., except that two LUTs are constructed rather than one, and one LUT or the other is used depending on whether the apparent fast axis orientation is approximately the same, or approximately orthogonal, at the two wavelengths. The former LUT table is populated along lines having slope z≈+1 while the other is populated along lines having z≈xe2x88x921. There are no intersections or crossed lines on either LUT, nor any case where two different R values correspond to a given set of observed {R0(xcex1), R0(xcex2)}. Thus, there is no ambiguity in deriving R correctly.
Additional features include using empirical knowledge of the signal-to-noise in the system to describe the maximum possible error in R0 at each wavelength, from which one can determine that certain combinations of {R0(xcex1), R0(xcex2)} are possible, while others cannot occur. This amounts to demarcating certain bands in the LUT as valid, and given values for R, while others are simply marked as invalid. The bands are different for the LUT where the fast axes are coincident, and the LUT where they are different. If the LUT indicates that the measured combination is invalid, the data may be marked as no good. This provides a measurement integrity check on the system.